Delocalization effect of the Hubbard repulsion in exact terms and two dimensions

The genuine physical reasons explaining the delocalization effect of the Hubbard repulsion $U$ are analyzed. First, it is shown that always when this effect is observed, $U$ acts on the background of a macroscopic degeneracy present in a multiband type of system. After this step, I demonstrate that acting in such conditions, by strongly diminishing the double occupancy, $U$ spreads out the contributions in the ground state wave function, hence strongly increases the one-particle localization length, and consequently extends the one-particle behavior producing conditions for a delocalization effect. To be valuable, the reported results are presented in exact terms, being based on the first exact ground states deduced at half-filling in two dimensions for a prototype two band system, which is the generic periodic Anderson model at finite value of the interaction.

Figure 1

The cell defined at site $\mathbf{i}$ (thick line) containing five sites of the lattice $\left(\mathbf{i},\mathbf{i}\pm \mathbf{x},\mathbf{i}\pm \mathbf{y}\right)$ on which the operator ${\widehat{A}}_{\mathbf{i},v,e}$ is defined. The sites of the sublattice $v$, $\mathbf{i}\in S_v$, are denoted by black dots, $\left(\mathbf{x},\mathbf{y}\right)$ are the primitive vectors of the lattice, and $n=1,2,\dots ,5$ represents the notation of sites inside the cell.

Figure 2

An $8\times 8$ lattice (e.g., $L=8$ holds) containing one straight line $l_{\mathbf{y}+\mathbf{x},v=1,n}$ (thick line). The direction of the line is $\mathbf{x}+\mathbf{y}$, it is contained in the sublattice $v=1$ denoted by full circles, the line contains $L=8$ sites, and note that periodic boundary conditions are considered. $\left(\mathbf{x},\mathbf{y}\right)$ are the primitive vectors of the 2D Bravais lattice. In the $\mathbf{x}+\mathbf{y}$ direction $L/2=4$, such lines can be plotted.

Figure 3

The four possible breaking directions denoted by the index $\tau$.

Figure 4

A broken line $l_v\left(\mathbf{i},{\tau }_{\mathbf{i}}=1\right)$ (thick line) defined at the site $\mathbf{i}$ in a $8\times 8$ lattice. $l_v\left(\mathbf{i},{\tau }_{\mathbf{i}}=1\right)$ connects sites of the $v$ sublattice (full dots). Note that a number of $L^2/2$ such broken lines can be plotted for a fixed $v$.

Figure 5

The first cusp region of ${\widehat{W}}_{v=1,\mathbf{i},\tau =1,{\lambda }_v,e_v}^{†}$ constructed along the broken line (thick line), which is labeled by $\left(\mathbf{i},{\tau }_{\mathbf{i}}=1\right)$. White, gray, and full circles represent in order ${\widehat{D}}^{†}$, ${\widehat{F}}^{†}$, and ${\widehat{P}}^{†}$ operators; the coefficients ($x_l,y_l,ϵ$, etc.) are the numerical prefactors.

Figure 6

The second cusp region of ${\widehat{W}}_{v=1,\mathbf{i},\tau =1,{\lambda }_v,e_v}^{†}$. The thick line is a part of the broken line. For notations, see Fig. 5.

Figure 7

Intercusps (straight) region of ${\widehat{W}}_{v=1,\mathbf{i},\tau =1,{\lambda }_v,e_v}^{†}$. The thick line is a part of the broken line. For notations, see Fig. 5.

Figure 8

Ground state expectation value of $f$-hopping correlations $\Gamma \left(\mathbf{r}\right)$ at $U=0$ (dashed line) and $U\ne 0$ (continuous line) for a $12\times 12$ system at $t_1/t_2=10,t_2/V_1=0.5,V_1/V_0=0.1$. $\mathbf{r}\parallel \left(\mathbf{x}+\mathbf{y}\right)$ is in $\left|\mathbf{x}+\mathbf{y}\right|$ units. The inset shows the log-log plot at $U\ne 0$. For example, at $\mathbf{r}=4$, ${\Gamma }_U/{\Gamma }_{U=0}=2\times {10}^5$ holds (Ref. 28). The result is qualitatively the same for other directions and other $V_1/V_0$ ratios as well.